Translation-invariant linear forms on \({\mathcal E}'(G)\) and non-Liouville tuples on G
type: Item
instance of: scholarly article
title: Translation-invariant linear forms on \({\mathcal E}'(G)\) and non-Liouville tuples on G
zbMATH Open document ID: 0649.43001
DOI: 10.1016/0022-1236(88)90128-0
author: Lilian Asam
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let G be a locally compact, compactly generated Abelian group; let \({\mathcal E}'(G)\) be the Schwartz-Bruhat space of distributions on G having compact supports. Then every translation-invariant linear form on \({\mathcal E}'(G)\) is continuous if and only if G contains a finite ``non-Liouville'' set. To understand this, one must have a definition of ``non-Liouville set''; this, like the theorem itself, is a generalization from the case of the torus group. If \(x\in {\mathbb{R}}\), let \(x\) denote the minimal distance from x to an integer. Let \(\Pi\) denote the canonical map from \({\mathbb{R}}^ m\) onto \({\mathbb{T}}^ m={\mathbb{R}}^ m/{\mathbb{Z}}^ m\). A subset \(\{t_ 1,...,t_ n\}\) of \({\mathbb{T}}^ m\) is a non-Liouville n- tuple if and only if there are real numbers C, \(b>0\) such that, for any \(a_{\nu}\) satisfying \(\Pi (a_{\nu})=t_{\nu}\), and any nonzero \(k\in {\mathbb{Z}}^ m\), \(\sum^{n}_{\nu =1}[[ <k,a_{\nu}>]]\geq C(1+| k|)^{-b}\). An equivalent condition is: for every character \(\chi\) of the group \({\mathbb{T}}^ m\), not identically equal to 1, \(\sum^{n}_{\nu =1}| 1-\chi (a_{\nu})| \geq C(1+| k|)^{-b}\) (for a possibly different value of C). The latter inequality is easily generalized from \({\mathbb{T}}^ m\) to an arbitrary compact Abelian group G and its dual. If G is a locally compact Abelian group, then a finite subset \(\{a_ 1,...,a_ n\}\) is a non-Liouville n-tuple iff there is a discrete, finitely generated subgroup H of G such that G/H is compact and the images of \(a_ 1,...,a_ n\) are a non-Liouville n-tuple in G/H.
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 46F99
zbMATH DE Number: 4060049
zbMATH Keywords: locally compact, compactly generated Abelian group
zbMATH Keywords: Schwartz-Bruhat space
zbMATH Keywords: distributions
zbMATH Keywords: compact supports
zbMATH Keywords: translation-invariant linear form
zbMATH Keywords: non-Liouville n-tuple
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90128-0
OpenAlex ID: W2030049734
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description: scientific article; zbMATH DE number 4060049